1.3 Errors and uncertainties
- Syllabus
- 9702–2028–2029
- Topic
- 1.3
- Level
- AS
Learning objectives
1.3.1The effects of systematic errors (including zero errors) and random errors• understand and explain the effects of systematic errors (including zero errors) and random errors in measurements1.3.2The distinction between precision and accuracy• understand the distinction between precision and accuracy1.3.3Assess the uncertainty in a derived quantity by simple addition of absolute• assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties
Systematic error shifts results; random error scatters them
A systematic error biases measurements in a consistent way, so results tend to be too high or too low. A random error causes unpredictable variation, producing scatter among repeated readings.
Repeating and averaging readings reduces the effect of random error. It does not remove systematic error; that requires a zero check, calibration, correction or a changed method.
A zero error is systematic: if a balance reads +0.20 g when empty, its uncorrected readings are shifted upward by 0.20 g. By contrast, reading an analogue scale from slightly different angles can make readings scatter.
A systematic error need not add the same fixed amount; faulty calibration may make the bias vary with the reading. The defining feature is consistent bias, not simply a constant offset.
Precision is agreement; accuracy is closeness to the true value
Precision describes how closely repeated measurements agree with one another. Accuracy describes how close a measured value is to the true or accepted value.
Judge precision from the spread or uncertainty of repeated readings. Judge accuracy by comparison with a true or accepted value, or by evidence of systematic bias.
If the true value is 10.8 cm, readings 10.1 cm, 10.1 cm and 10.2 cm are precise because they cluster tightly, but inaccurate because the cluster is far from 10.8 cm.
Extra decimal places show resolution, not guaranteed precision or accuracy. A tight cluster may still be systematically shifted from the true value.
The operation determines how uncertainties combine
For a sum or difference, add absolute uncertainties. For a product or quotient, add percentage uncertainties. If a measured quantity is raised to a power, multiply its percentage uncertainty by the magnitude of that power.
| Derived form | Uncertainty rule |
|---|---|
| Q = x + y or x − y | ΔQ = Δx + Δy |
| Q = xy or x/y | %ΔQ = %Δx + %Δy |
| Q = xⁿ | %ΔQ = |
A temperature rise is (100.0 ± 0.5) °C − (40.0 ± 0.5) °C. The rise is 60.0 °C and its absolute uncertainty is 0.5 + 0.5 = 1.0 °C, so the result is (60.0 ± 1.0) °C. Its percentage uncertainty is (1.0/60.0) × 100% = 1.7%.
Choose the rule from the mathematical operation, not from the units. Exact numerical constants do not contribute measurement uncertainty; a power changes the percentage contribution of the measured quantity.